Let , or in general finite dimensional -vector space. Then we define (or ) as the set of all -dimensional subspaces (which are referred to as “-planes”) of . (Of course, we assume .)
Any -plane may be represented by an matrix of rank . That is, the rows of the matrix are basis elements of the -plane . Denote the set of all matrices with rank . Then we may identify
, the orbit of group action by .
Exercise 1.1. Justify the above identification by showing the following statement. Any matrices and represent same -planes if and only if there exists such that .
(Hint: Show that represent the same -plane if and only if there is matrix such that for each , where denote the rows of , respectively. Let and denote the th columns of and respectively. Take the transpose to get . Thus represent the same -planes if and only if there are matrices such that and . Show that so that .)
Exercise 1.2. Show that .
(Hint: From the hint given for Exercise 1.1., show that for any matrices , the row spaces of coincide if and only if the column spaces of them coincide. Then for any rank matrix , we may assume that is given by augmenting an matrix on the right to a invertible matrix (i.e., an element of ). Argue that the entries of decide the dimension of the whole grassmannian.)
Recall that we can projectivize a vector space as follows:
If then clearly . We have the identification .
Example. We have
There is another way to describe . We review the relevant linear algebra first.
Recall that is the set of alternating -multilinear functions . Given and , we define
That is, we have
Exercise 1.3. Show that .
Exercise 1.4. Let be the standard dual basis. Given and , show that , which is when and otherwise. Conclude that the set form a basis for . (In particular, we have , and of course we are assuming . If , exterior -power is just zero.)
Convention. Given , we have a unique dual image . Thus, if , then we write .
Exercise 1.5. For the standard basis , show that .
Exercise 1.6. Given the bases , take the matrix given by . In other words, in basis the th row of are . Show that
Thus if is an matrix, then
Now we describe in a different way. Fix any . Consider any basis . Then we have a map . If we write and , then is the matrix whose rows are . Writing , we have . Thus for any other basis , we have
Now define by choosing a basis since is not dependent on the choice of basis of .
More concretely, if we write . Then , where . This means that we can recognize as the space of skew-symmetric matrices:
Since , we have
The following exercise gives an intuitive picture of .
Exercise 1.7. Given , define . Given , show .
Exercise 1.8. Given an skew symmetric matrix , if is odd, then .
Exercise 1.9. (Hard) Given an skew symmetric matrix , if is even and , then there is a polynomial such that . This polynomial is called the Pfaffian of .
Exercise 1.10. Explain how to define the rank of elements in (Hint: use skew symmetric matrix).